Non-Bloch bands in two-dimensional non-Hermitian systems Kazuki Yokomizo1and Shuichi Murakami2 1Department of Physics The University of Tokyo

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Non-Bloch bands in two-dimensional non-Hermitian systems

Kazuki Yokomizo1and Shuichi Murakami2

1Department of Physics, The University of Tokyo,

7-3-1 Hongo, Bunkyo-ku, Tokyo, 113-0033, Japan

2Department of Physics, Tokyo Institute of Technology,

2-12-1 Ookayama, Meguro-ku, Tokyo, 152-8551, Japan

The non-Bloch band theory can describe energy bands in a one-dimensional (1D) non-Hermitian

system. On the other hand, whether the non-Bloch band theory can be extended to higher-

dimensional non-Hermitian systems is nontrivial. In this work, we construct the non-Bloch band

theory in two classes of two-dimensional non-Hermitian systems, by reducing the problem to that

for a 1D non-Hermitian model. In these classes of systems, we get the generalized Brillouin zone

for a complex wavevector and investigate topological properties. In the model of the non-Hermitian

Chern insulator, as an example, we show the bulk-edge correspondence between the Chern number

deﬁned from the generalized Brillouin zone and the appearance of the edge states.

I. INTRODUCTION

A non-Hermitian Hamiltonian eﬀectively describes a

nonequilibrium system [1]. In recent studies, the non-

Hermitian skin eﬀect, i.e., the localization of bulk eigen-

states, plays a crucial role because it causes remarkable

phenomena [2–24]. In particular, with the non-Hermitian

skin eﬀect, energy eigenvalues under an open boundary

condition and those under a periodic boundary condi-

tion are diﬀerent. Such phenomena associated with the

non-Hermitian skin eﬀect are an obstacle to investigate

properties of a non-Hermitian system. For example, the

bulk-edge correspondence between a topological invari-

ant deﬁned from the conventional Bloch wavevector and

existence of topological edge states seems to be violated

within the conventional deﬁnition of a topological invari-

ant.

Recently, the non-Bloch band theory was proposed [2,

25–28]. It was shown that in a one-dimensional (1D)

spatially periodic non-Hermitian system without disor-

der, the generalized Brillouin zone β=eik for the com-

plex Bloch wavenumber kbecomes diﬀerent from that

in a Hermitian system, and it reproduces energy eigen-

values under an open boundary condition. The non-

Bloch band theory gives the condition for the general-

ized Brillouin zone and is applicable to the system even

with long-range hopping amplitudes and on-site poten-

tials. Importantly, a topological invariant deﬁned from

the generalized Brillouin zone restores the bulk-edge cor-

respondence. Additionally, some features of the gener-

alized Brillouin zone lead to unique phenomena, such as

appearance of a topological semimetal phase with excep-

tional points [29]. Thus, the non-Bloch band theory is

a powerful tool for studies on many properties in a non-

Hermitian system. However, there is a lack of studies on

a two-dimensional (2D) non-Hermitian system in terms

of the non-Bloch band theory.

There have been some studies on topological phases

and boundary states in a 2D non-Hermitian system

in terms of the conventional Bloch band theory. For

example, some previous works investigated how non-

Hermiticity aﬀects the bulk-edge correspondence [30–41].

Furthermore, Refs. [42–45] calculated the Hall conduc-

tance associated with the topological edge states in the

model of the non-Hermitian Chern insulator. In recent

years, non-Hermitian topology in a higher-dimensional

system has been attracting much attention. In the-

ory, Refs. [46–53] proposed the higher-order topological

non-Hermitian skin eﬀect, in which eigenstates are lo-

calized at corners of the system, and in experiment, it

has been observed [54–58]. We emphasize that the topo-

logical invariant predicting the higher-order topological

non-Hermitian skin eﬀect is deﬁned from the real Bloch

wavevector. However, as we learned from a 1D non-

Hermitian system, we cannot investigate physics of a 2D

non-Hermitian system with an open boundary condition

in terms of the real Bloch wavevector. Therefore, it is

necessary to construct the non-Bloch band theory in a

2D non-Hermitian system.

So far, the non-Bloch bands in some 2D non-Hermitian

models were investigated [59–65]. Nevertheless, it is

unclear whether the non-Bloch band theory in general

2D non-Hermitian systems can be constructed. In this

work, we construct the non-Bloch band theory in two

classes of 2D non-Hermitian systems, where we can re-

duce the problem to that of a 1D non-Hermitian model.

One class of systems has speciﬁc symmetries, such as

a mirror symmetry suppressing the non-Hermitian skin

eﬀect in one direction. The other class is decoupled

into two 1D non-Hermitian systems due to a special

form of the Hamiltonian. These properties indicate that

the eigenstates in the bulk are written as linear com-

bination of a few plane waves. Thereby, the condi-

tion for the generalized Brillouin zone can be obtained.

In this paper, we exemplify three systems, e.g., the

model of the non-Hermitian Chern insulator, the non-

Hermitian Benalcazar–Bernevig–Hughes (BBH) model,

and the Okugawa-Takahashi-Yokomizo (OTY) model.

We show that the generalized Brillouin zone reproduces

the energy eigenvalues in these models with a full open

boundary condition. Furthermore, we investigate topo-

logical properties in terms of the generalized Brillouin

zone.

arXiv:2210.04412v2 [cond-mat.mes-hall] 9 May 2023

FIG. 1. Two-dimensional tight-binding system on a square

lattice. The geometry of the system is a rectangle, where the

lengths in the xdirection and in the ydirection are Lxand

Ly, respectively. The black circles express the unit cells at the

lattice point r= (nx, ny) (nx= 1,...,Lx, ny= 1,...,Ly).

This paper is organized as follows: In Sec. II, we in-

troduce our 2D non-Hermitian tight-binding model and

propose two classes of systems where we can apply the

non-Bloch band theory. Then, we can get the condition

for the generalized Brillouin zone. In Sec. III, we ana-

lyze the three 2D non-Hermitian models in terms of the

non-Bloch band theory. In Sec. IV, we discuss the 2D

non-Bloch band theory proposed in this work from the

viewpoint of the formation of standing wave and com-

ment on diﬃculty to construct the non-Bloch band the-

ory in general 2D non-Hermitian systems. Finally, in

Sec. V, we summarize our result.

II. MODEL

First of all, we introduce a 2D non-Hermitian tight-

binding system. Throughout this paper, the system lies

on a square lattice, and the geometry of the system is a

rectangle, as shown in Fig. 1. The real-space Hamiltonian

of our system is given by

H=X

µ,ν=1 Nx

i=−Nx

t(x)

i,µν c†

r+iˆ

x,µcr,ν

i=−Ny

t(y)

i,µν c†

r+iˆ

y,µcr,ν 

,(1)

where c†

r,µ is a creation operator, r=

(nx, ny) (nx= 1, . . . , Lx, ny= 1, . . . , Ly) is a lat-

tice point, qis the number of internal degrees of freedom

in a unit cell, and ˆ

xand ˆ

yare unit vectors in the x

direction and in the ydirection, respectively. The parti-

cles hop up to the Nxth and Nyth nearest-neighbor unit

cells along the xand ydirections, respectively. We note

that when either t(x)

i,νµ 6=t(x)

−i,µν ∗

or t(y)

i,νµ 6=t(y)

−i,µν ∗

is satisﬁed, the system becomes non-Hermitian. Now,

the real-space eigen-equation is written as

H|ψi=E|ψi,(2)

where the eigenstates are given by

|ψi=...,ψ(nx,ny),1, . . . , ψ(nx,ny,1),q, . . . T.(3)

Next, we describe a way to construct the non-Bloch

band theory in our system. In the case of q= 1, it is

straightforward to construct the non-Bloch band theory

in the system. This is because the system can be de-

composed into two 1D non-Hermitian models. In Ap-

pendix A, we derive the condition for the generalized

Brillouin zone in this case. On the other hand, in the

case of q≥2, we ﬁnd that when the system satisﬁes

some conditions, one can calculate the generalized Bril-

louin zone in terms of the conventional non-Bloch band

theory. In the following, we explain the two cases where

the description of the non-Bloch bands is possible.

A. Case A

In a 1D non-Hermitian system, when the energy of

the eigenstate with the Bloch wavenumber +kand that

of the eigenstate with −kare degenerate, the non-

Hermitian skin eﬀect is suppressed because kbecomes

real [8, 66, 67]. The degeneracy leading to the suppres-

sion of the non-Hermitian skin eﬀect comes from symme-

tries, e.g., a global symmetry, such as a PT symmetry,

and a crystalline symmetry, such as a mirror symmetry.

Importantly, we here propose that such suppression is re-

alized also in a 2D non-Hermitian system. For example,

when the Bloch Hamiltonian H(kx, ky) satisﬁes

U−1HT(kx,−ky)U=H(kx, ky),(4)

where Uis a unitary matrix satisfying U2= 1, we can

show that the non-Hermitian skin eﬀect in the ydirection

is suppressed. In fact, when Eq. (4) is satisﬁed, the wave

with the Bloch wavevector (kx, ky) is reﬂected to that

with (kx,−ky) at the boundary of the system parallel to

the xdirection. Then, the condition for the formation

of standing wave means that the imaginary parts of the

wavevector for the incident and reﬂected waves are equal.

Therefore, we can obtain Im (ky) = Im (−ky), meaning

that kyis real, and the non-Hermitian skin eﬀect in the

ydirection is suppressed.

In this case, the bulk eigenstates extend over the ydi-

rection because the Bloch wavenumber kybecomes real.

Hence, in the limit of a large system size, the asymptotic

behavior of the bulk eigenstates in the cylinder geometry,

where a periodic boundary condition in the ydirection

is imposed to the system, matches that in the full-open

geometry. Therefore, the energy bands in the cylinder

geometry asymptotically reproduces the energy levels in

a ﬁnite open plane. Importantly, since the cylinder ge-

ometry can be regarded as a pseudo-1D system, one can

calculate the energy bands of this system in terms of the

conventional non-Bloch band theory. Thus, we can con-

struct the non-Bloch band theory in the system.

Based on the above concept, we establish the non-

Bloch band theory in the 2D non-Hermitian systems

without the non-Hermitian skin eﬀect in the ydirection.

With Eq. (4), by assuming that the Bloch wavenumber

kyis real, we take

ψr,µ =X

(βx,j )nxeikynyφ(j)

µ(5)

as an ansatz for the eigen-equation in the cylinder geom-

etry. Here, we deﬁne the non-Bloch matrix as

[H(βx, ky)]µν =

m=−Nx

t(x)

m,µν (βx)m+

m=−Ny

t(y)

m,µν eikym.

(6)

Then, in Eq. (5), βx=βx,j is the solution of the charac-

teristic equation

det [H(βx, ky)−E]=0.(7)

We note that Eq. (7) is an algebraic equation for βxof

2qNxth degree. Therefore, the condition for the general-

ized Brillouin zone is given by

|βx,qNx|=|βx,qNx+1|,(8)

where the solutions of Eq. (7) are numbered as

|βx,1| ≤ · · · ≤ |βx,2qNx|.(9)

For a given real value of ky, we can obtain trajectories of

βx,qNxand βx,qNx+1 with Eq. (8) forming loops on the

complex plane. Thus, by changing the real value of ky,

the surface formed by (βx, βy)βy≡eiky, ky∈Ris the

generalized Brillouin zone. Finally, from the generalized

Brillouin zone, we can calculate the energy bands by us-

ing Eq. (7). In this work, we show that it matches the

energy levels in a ﬁnite open plane. This means that the

non-Hermitian skin eﬀect is indeed suppressed in the y

direction in this case.

B. Case B

Next, we focus on the case that the characteristic equa-

tion of the Bloch Hamiltonian H(kx, ky) is written in the

form of separation of variables. In this case, the motion

of the bulk eigenstates in the xdirection is completely

decoupled from that in the ydirection. Namely, the 2D

non-Hermitian systems can be regarded as two 1D non-

Hermitian systems. In the following, we assume that the

characteristic equation is given by

det [H(kx, ky)−E] = Rxeikx, E+Ryeiky= 0.

(10)

Then, by applying the conventional non-Bloch band the-

ory to the 1D non-Hermitian systems, we expect that

one can get the energy bands in the 2D non-Hermitian

system, and indeed, we will show it in this work.

Now, we derive the condition for the generalized Bril-

louin zone to calculate the energy bands. With Eq. (10),

we have

ψr,µ =X

jx,jy

(βx,jx)nxβy,jynyφ(jx,jy)

µ(11)

as an ansatz for Eq. (2). Here, βx=βx.jxand βy=βy.jy

in Eq. (11) are the solutions of the equation Rx(βx, E) =

λand the equation Ry(βy) = −λ, respectively, where λis

an arbitrary constant. We note that these equations are

algebraic equations of 2qNxth degree and 2qNyth degree,

respectively. Thus, we can get the generalized Brillouin

zone spanned by (βx, βy) satisfying

|βa,qNa|=|βa,qNa+1|,(12)

where the solutions of Rx(βx, E) = λand Ry(βy) = −λ

are numbered as

|βa,1| ≤ · · · ≤ |βa,2qNa|(13)

for a=x, y. Finally, the energy bands are obtained from

the characteristic equation of the non-Bloch matrix

[H(βx, βy)]µν =

m=−Nx

t(x)

m,µν (βx)m+

m=−Ny

t(y)

m,µν (βy)m.

(14)

We note that unlike Case A, the non-Hermitian skin ef-

fect appear in both directions.

III. EXAMPLES

In this section, we calculate the generalized Brillouin

zone and the energy bands in some examples, e.g., the

model of the non-Hermitian Chern insulator, the non-

Hermitian BBH model, and the OTY model. The ﬁrst

model is in Case A, and the second and third models are

in Case B, as discussed in Sec. II. Throughout this cal-

culation, we show that the non-Bloch band theory in the

Case A and Case B indeed reproduces the energy levels

in a ﬁnite open plane. Furthermore, we study topological

properties in theses systems in terms of the generalized

Brillouin zone.

A. Non-Hermitian Chern insulator

First of all, we focus on the model of a non-Hermitian

Chern insulator investigated in Ref. [59]. The real-space

Hamiltonian of this system is given by

H=X

rc†

r+ˆ

xT†

xcr+c†

rTxcr+ˆ

+c†

r+ˆ

yT†

ycr+c†

rTycr+ˆ

y+r†

nMcr,(15)

where

Tx=1

2−tx−ivx

−ivxtx, Ty=1

2−ty−vy

vyty

M=m iγx+γy

iγx−γy−m,(16)

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Non-Blochbandsintwo-dimensionalnon-HermitiansystemsKazukiYokomizo1andShuichiMurakami21DepartmentofPhysics,TheUniversityofTokyo,7-3-1Hongo,Bunkyo-ku,Tokyo,113-0033,Japan2DepartmentofPhysics,TokyoInstituteofTechnology,2-12-1Ookayama,Meguro-ku,Tokyo,152-8551,JapanThenon-Blochbandtheorycandescribeenergy...

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Non-Bloch bands in two-dimensional non-Hermitian systems Kazuki Yokomizo1and Shuichi Murakami2 1Department of Physics The University of Tokyo

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